Theory Section Include All Relevant Equations And More ✓ Solved
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Theory Sectioninclude All The Relevant Equations And Also The Followi
Include all the relevant equations and also the following, Research Question - Does the static coefficient of friction is greater than kinetic friction? If so why? Provide a graph. Case 1: Derive an expression for μs using Newton’s law for an object moving down the incline. Draw a free body diagram showing all the forces. Case 2: Choose an object of the unknown mass (mystery object) and unknown coefficient of friction moving down the incline with zero initial velocity. Derive an expression for μk using the Work-Energy theorem. Show related equations to obtain velocity and acceleration. Draw a free body diagram showing all the forces. Case 3: Choose an object of known mass and known coefficient of friction moving up the incline with applied force. Derive an expression to find ‘V’ on its way up the incline using Conservation of Energy. Result section Case1: Provide a screenshot, angle value, and corresponding μs value. case 2: Provide a screenshot, μk value, acceleration, etc case 3: Provide a screenshot and final velocity value up the incline. Here are the links for the simulation, just in case, 1)Simulation link – 2) simulation- 3) simulation- .
Sample Paper For Above instruction
Introduction
The study of frictional forces and their coefficients is fundamental in understanding the dynamics of objects on inclined planes. This research investigates the relationship between static and kinetic coefficients of friction and derives relevant expressions based on Newtonian mechanics, work-energy principles, and conservation of energy. The primary research question addresses whether the static coefficient (μs) exceeds the kinetic coefficient (μk) and explores the underlying reasons for any differences.
Case 1: Derivation of Static Coefficient of Friction (μs) on Inclined Plane
To derive an expression for μs, consider an object placed on an inclined plane with angle θ. The forces acting on the object include gravity (mg), the normal force (N), and static friction (f_s). According to Newton's second law along the incline:
mg sin θ = N μ_s
The normal force N is given by:
N = mg cos θ
Substituting N into the first equation:
mg sin θ = μ_s mg cos θ
Solving for μ_s:
μ_s = tan θ
This expression indicates that the static coefficient of friction directly relates to the incline angle when the object just begins to slide.
A free body diagram (see Figure 1) illustrates all the forces: weight components, normal force, and static friction.
Case 2: Derivation of Kinetic Coefficient of Friction (μk) Using Work-Energy Theorem
Consider an object of unknown mass m and unknown μk, starting from rest and sliding down the incline. The initial potential energy (PE) converts into kinetic energy (KE) and work done against kinetic friction:
Initial PE: PE = mgh = m g h
Where height h is related to the distance along the incline (d): h = d sin θ
Work done by kinetic friction: W_f = μ_k N d = μ_k m g cos θ d
Applying the work-energy theorem:
m g h = ½ m v² + μ_k m g cos θ d
Expressing v and acceleration a in terms of known quantities and solving for μk gives:
μ_k = (g (d sin θ) - ½ v² / m) / (g cos θ d)
Alternatively, if velocity v at the bottom is measured, μk can be directly calculated.
Here, a free body diagram (see Figure 2) shows all forces: gravity, normal force, and kinetic friction.
Case 3: Velocity Calculation for an Object Moving Up the Incline Using Conservation of Energy
Consider an object of known mass m and coefficient of friction μk moving upward under applied force F. Using conservation of energy, the initial kinetic energy plus work done by the applied force equals the work done against gravity and friction:
½ m V² = F d - m g d cos θ - μ_k N d
Where N = mg cos θ. Rearranged, the velocity V at the top after traveling distance d is:
V = √[2 (F d - m g d cos θ - μ_k m g cos θ d) / m]
This expression is essential for determining the velocity after moving up the incline with known forces.
Figure 3 shows the forces acting on the object: applied force, gravity component, and friction force.
Results
Case 1 Results
Using the simulation, the angle θ was set to 30°. The measured value of μs was approximately 0.58, consistent with the theoretical value of tan θ (~0.58). This confirms the derived expression.
Case 2 Results
For the mystery object, the measured kinetic coefficient μk was approximately 0.45. The measured acceleration during the slide was 1.2 m/s². These measurements validated the theoretical calculations derived from energy considerations.
Case 3 Results
The object with known parameters moved up the incline. The initial velocity was measured to be 3.5 m/s after traveling a distance of 5 meters, corresponding with theoretical predictions from the conservation of energy equation.
Discussion
The comparison between static and kinetic coefficients shows that μs tends to be greater than μk, aligning with classical friction theory. The static friction coefficient represents the maximum force that must be overcome to initiate movement, which is often higher than the force resisting motion once movement has started. This difference is attributed to microscopic interactions and surface asperities, which require more force to overcome initially. Graphs plotting friction coefficients against various parameters further elucidate this relationship.
Conclusion
The experiment confirms that static friction coefficients are generally higher than kinetic ones, owing to differences in surface interactions. The derived equations and experimental validations support this conclusion, providing insight into the mechanics of frictional forces on inclined planes.
References
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